A036968 -id:A036968 - cp's OEIS frontend

A239275 a(n) = numerator(2^n * Bernoulli(n, 1)).

1, 1, 2, 0, -8, 0, 32, 0, -128, 0, 2560, 0, -1415168, 0, 57344, 0, -118521856, 0, 5749735424, 0, -91546451968, 0, 1792043646976, 0, -1982765704675328, 0, 286994513002496, 0, -3187598700536922112, 0, 4625594563496048066560, 0, -16555640873195841519616, 0, 22142170101965089931264, 0

Offset: 0

Paul Curtz, Mar 13 2014

Difference table of f(n) = 2^n *A164555(n)/A027642(n) = a(n)/A141459(n):
  1,           1,      2/3,        0,    -8/15,      0,  32/21, 0,...
  0,        -1/3,     -2/3,    -8/15,     8/15,  32/21, -32/21,...
  -1/3,     -1/3,     2/15,    16/15,  104/105, -64/21,...
  0,        7/15,    14/15,   -8/105, -424/105,...
  7/15,     7/15, -106/105, -416/105,...
  0,      -31/21,   -62/31,
  -31/21, -31/21,...
  0,... etc.
Main diagonal: A212196(n)/A181131(n). See A190339(n).
First upper diagonal: A229023(n)/A181131(n).
The inverse binomial transform of f(n) is g(n). Reciprocally, the inverse binomial transform of g(n) is f(n) with -1 instead of f(1)=1, i.e., f(n) signed.
Sum of the antidiagonals: 1,1,0,-1,0,3,0,-17,... = (-1)^n*A036968(n) = -A226158(n+1).
Following A211163(n+2), f(n) is the coefficients of a polynomial in Pi^n.
Bernoulli numbers, twice, and Genocchi numbers, twice, are linked to Pi.
f(n) - g(n) = -A226158(n).
Also the numerators of the centralized Bernoulli polynomials 2^n*Bernoulli(n, x/2+1/2) evaluated at x=1. The denominators are A141459. - Peter Luschny, Nov 22 2015
(-1)^n*a(n) = 2^n*numerator(A027641(n)/A027642(n)) (that is the present sequence with a(1) = -1 instead of +1). - Wolfdieter Lang, Jul 05 2017

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Wolfdieter Lang, On Sums of Powers of Arithmetic Progressions, and Generalized Stirling, Eulerian and Bernoulli Numbers, arXiv:math/1707.04451 [math.NT], July 2017. See B(2;n), eq. (53).

Cf. A141459 (denominators), A001896/A001897, A027641/A027642.

seq(numer(2^n*bernoulli(n, 1)), n=0..35); # Peter Luschny, Jul 17 2017

Table[Numerator[2^n*BernoulliB[n, 1]], {n, 0, 100}] (* Indranil Ghosh, Jul 18 2017 *)

from sympy import bernoulli
def a(n): return (2**n * bernoulli(n, 1)).numerator
print([a(n) for n in range(51)]) # Indranil Ghosh, Jul 18 2017

a(n) = numerators of 2^n * A164555(n)/A027642(n).

Numerators of the binomial transform of A157779(n)/(interleave A001897(n), 1)(conjectured).

A227608 Denominators of A225825(n) difference table written by antidiagonals.

Original entry on oeis.org

1, 2, 2, 6, 3, 6, 2, 3, 3, 2, 30, 15, 15, 15, 30, 2, 15, 15, 15, 15, 2, 42, 21, 105, 105, 105, 21, 42, 2, 21, 21, 105, 105, 21, 21, 2, 30, 15, 105, 105, 105, 105, 105, 15, 30, 2, 15, 15, 105, 105, 105, 105, 15, 15, 2, 66, 33, 165, 165, 1155, 231, 1155, 165, 165, 33, 66, 2, 33, 33, 165, 165, 231, 231, 165, 165, 33, 33, 2

Offset: 0

Paul Curtz, Aug 10 2013

			1,
-1/2,      1/2,
-1/6,     -2/3,     -1/6,
1/2,       1/3,     -1/3,     -1/2,
7/30,    11/15,    16/15,    11/15,     7/30,
-3/2,   -19/15,    -8/15,     8/15,    19/15,    3/2,
-31/42, -47/21, -368/105, -424/105, -368/105, -47/21, -31/42.
Row sums: 1, 0/2, -6/6, 0/6, 90/30, 0/30, -3570/210, 0/210, 32550/210,... .
Are the denominators A034386(n+1)?
Reduced row sums: 1, 0, -1, 0, 3, 0, -17, 0, 155,... = -A036968(n+1)? See A226158(n+2). First 100 terms checked by Jean-François Alcover.

Cf. A085738

max = 12; b[0] = 1; b[n_] := Numerator[ BernoulliB[n, 1/2] - (n+1)*EulerE[n, 0]]; t = Table[b[n], {n, 0, max}] / Table[ Sum[ Boole[ PrimeQ[d+1]]/(d+1), {d, Divisors[n]}] // Denominator, {n, 0, max}]; dt = Table[ Differences[t, n], {n, 0, max}]; Table[ dt[[n-k+1, k]] // Denominator, {n, 1, max}, {k, 1, n}] // Flatten (* Jean-François Alcover, Aug 12 2013 *)

More terms from Jean-François Alcover, Aug 12 2013

A321217 Genocchi irregular primes.

Original entry on oeis.org

17, 31, 37, 41, 43, 59, 67, 73, 89, 97, 101, 103, 109, 113, 127, 131, 137, 149, 151, 157, 193, 223, 229, 233, 241, 251, 257, 263, 271, 277, 281, 283, 293, 307, 311, 313, 331, 337, 347, 353, 379, 389, 397, 401, 409, 421, 431, 433, 439, 449, 457, 461, 463, 467, 491, 499

Offset: 1

Michel Marcus, Oct 31 2018

nonn

An odd prime p is G-irregular if it divides at least one of the integers G2, G4, ..., G(p-3).

Conjecture (Hu et al., 2019): The asymptotic density of this sequence within the primes is 1 - 3*A/(2*sqrt(e)) = 0.659776..., where A is Artin's constant (A005596). - Amiram Eldar, Dec 06 2022

Amiram Eldar, Table of n, a(n) for n = 1..1024
Su Hu, Min-Soo Kim, Pieter Moree, and Min Sha, Irregular primes with respect to Genocchi numbers and Artin's primitive root conjecture, Journal of Number Theory, Vol. 205 (2019), pp. 59-80, preprint, arXiv:1809.08431 [math.NT], 2019.
Peter Luschny, Irregular Bernoulli and Euler Primes.
Pieter Moree and Min Sha, Primes in arithmetic progressions and nonprimitive roots, Bulletin of the Australian Mathematical Society (2019), preprint, arXiv:1901.02650 [math.NT], 2019.
Pieter Moree and Pietro Sgobba, Prime divisors of l-Genocchi numbers and the ubiquity of Ramanujan-style congruences of level l, arXiv:2209.08047 [math.NT], 2022.

Cf. A036968 (Genocchi numbers), A000928 (irregular primes), A120337 (Euler-irregular primes), A128197 (strong irregular primes), A250216 (weak irregular primes), A005596.

A321217_list := proc(bound)
   local ae, F, p, m, maxp; F := NULL;
   for m from 2 by 2 to bound do
      p := nextprime(m+1);
      ae := abs(m*euler(m-1, 0));
      maxp := min(ae, bound);
      while p <= maxp do
          if ae mod p = 0 then F := F, p fi;
          p := nextprime(p)
      od
   od;
sort({F}) end: A321217_list(500); # Peter Luschny, Nov 11 2018

G[n_] := G[n] = n EulerE[n - 1, 0];
GenocchiIrregularQ[p_] := AnyTrue[Table[G[k], {k, 2, p-3, 2}], Divisible[#, p]&];
Select[Prime[Range[2, 100]], GenocchiIrregularQ] (* Jean-François Alcover, Nov 16 2018 *)

More terms from Peter Luschny, Nov 11 2018

A357240 Expansion of e.g.f. 2 * (exp(x) - 1) / (exp(exp(x) - 1) + 1).

Original entry on oeis.org

0, 1, 0, -2, -5, -4, 32, 225, 794, 190, -22291, -200298, -920244, 924223, 65848880, 716920754, 3831260555, -13147083976, -575844827780, -7162425813919, -40755845041730, 320194436283162, 11810647258173653, 161108090793013130, 896865861205240824, -14305712791762925929, -487306962045115504436

Offset: 0

Ilya Gutkovskiy, Sep 19 2022

sign

Stirling transform of the Genocchi numbers (of first kind, A036968).

Alois P. Heinz, Table of n, a(n) for n = 0..494

Cf. A001469, A003149, A036968, A059371.

b:= proc(n, m) option remember; `if`(n=0, `if`(m=0, 0,
      m*euler(m-1, 0)), m*b(n-1, m)+b(n-1, m+1))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..26);  # Alois P. Heinz, Jun 23 2023

nmax = 26; CoefficientList[Series[2 (Exp[x] - 1)/(Exp[Exp[x] - 1] + 1), {x, 0, nmax}], x] Range[0, nmax]!
Table[2 Sum[StirlingS2[n, k] (1 - 2^k) BernoulliB[k], {k, 0, n}], {n, 0, 26}]

a(n) = 2*sum(k=0, n, stirling(n, k, 2)*(1-2^k)*bernfrac(k)); \\ Michel Marcus, Sep 20 2022

a(n) = 2 * Sum_{k=0..n} Stirling2(n,k) * (1 - 2^k) * Bernoulli(k).

a(n) ~ Pi^(3/2) * 2^(n + 7/2) * n^(n + 1/2) * (cos(n*arctan(2*arctan(Pi)/log(1 + Pi^2))) * (Pi*log(1 + Pi^2) + 2*arctan(Pi)) + (log(1 + Pi^2) - 2*Pi*arctan(Pi)) * sin(n*arctan(2*arctan(Pi)/log(1 + Pi^2)))) / ((1 + Pi^2) * exp(n) * (4*arctan(Pi)^2 + log(1 + Pi^2)^2)^(n/2 + 1)). - Vaclav Kotesovec, Oct 04 2022

A278331 Shifted sequence of second differences of Genocchi numbers.

Original entry on oeis.org

0, -2, -2, 6, 14, -34, -138, 310, 1918, -4146, -36154, 76454, 891342, -1859138, -27891050, 57641238, 1080832286, -2219305810, -50833628826, 103886563462, 2853207760750, -5810302084962, -188424521441482, 382659344967926, 14464296482284734, -29311252309537394, -1277229462293249018

Offset: 0

Jean-François Alcover and Paul Curtz, Nov 18 2016

sign

This is an autosequence of the first kind (array of successive differences shows typical zero diagonal).
Last digits are apparently of period 20.
From A226158(n) for the continuity of autosequences of the first kind.
b(n) = 0, 1, -1, 0, 1, 0, -3, 0, 17, ... = A226158(n) with 1 as second term instead of -1.
c(n) = 0, 0, -1, 0, 1, 0, -3, 0, 17, ... = A226158(n) with 0 as second term instead of -1.
Respective difference tables:
0, -1, -1,  0,  1,  0, -3,   0,   17, ...
-1, 0,  1,  1, -1, -3 , 3,  17,  -17, ...
1,  1,  0, -2, -2,  6, 14, -34, -138, ...
etc,
0,  1, -1,  0,  1,  0, -3,   0,   17, ... = 0 followed by A036968(n+1)
1, -2,  1,  1, -1, -3,  3,  17,  -17, ...
-3, 3,  0, -2, -2,  6, 14, -34, -138, ...
etc,
0,  0, -1,  0,  1,  0, -3,   0,   17, ...
0, -1,  1,  1, -1, -3,  3,  17,  -17, ...
-1, 2,  0, -2, -2,  6, 14, -34, -138, ...
etc.
Since it is in the three tables, a(n) is the core of the Genocchi numbers.

Eric Weisstein's MathWorld, Genocchi Number.
Wikipedia, Genocchi number

Cf. A001469, A014781, A036968, A005439 (a(n) second and third diagonals), A164555/A027642, A209308, A226158, A240581(n)/A239315(n) (core of Bernoulli numbers).

g[0] = 0; g[1] = -1; g[n_] := n*EulerE[n-1, 0]; G = Table[g[n], {n, 0, 30}]; Drop[Differences[G, 2], 2]
(* or, from Seidel's triangle A014781: *)
max = 26; T[1, 1] = 1; T[n_, k_] /; 1 <= k <= (n + 1)/2 := T[n, k] = If[EvenQ[n], Sum[T[n - 1, i], {i, k, max}], Sum[T[n - 1, i], {i, 1, k}]]; T[, ] = 0; a[n_] := With[{k = Floor[(n - 1)/2] + 1}, (-1)^k*T[n + 3, k]]; Table[a[n], {n, 0, max}]

a(n) = (n+2)*E(n+1, 0) - 2*(n+3)*E(n+2, 0) + (n+4)*E(n+3, 0), where E(n,x) is the n-th Euler polynomial.

a(n) = -2*(2^(n+2)-1)*B(n+2) + 4*(2^(n+3)-1)*B(n+3) - 2*(2^(n+4)-1)*B(n+4), where B(n) is the n-th Bernoulli number.

A290701 a(n) = Sum_{k=1..n-1} binomial(n, k)G_nG_{n-k} where G_n is the n-th Genocchi number (of the first kind).

Original entry on oeis.org

2, -6, 6, 10, -30, -42, 238, 306, -2790, -3410, 45606, 53898, -993902, -1146810, 27887070, 31605346, -979901046, -1095183522, 42166810390, 46605422010, -2181617832702, -2389390959626, 133636947954126, 145257552124050, -9566483624198150, -10331802314134002

Offset: 2

Seiichi Manyama, Aug 09 2017

sign

Seiichi Manyama, Table of n, a(n) for n = 2..549
Wikipedia, Genocchi number

Cf. A036968.

a(n) = 2*n*A036968(n-1) + 2*(n-1)*A036968(n).

A305711 Expansion of e.g.f. exp(2*x/(exp(x) + 1)).

Original entry on oeis.org

1, 1, 0, -2, -1, 11, 13, -111, -220, 1756, 5051, -39775, -153191, 1215345, 5952668, -48020714, -288569149, 2377190003, 17069110381, -143857868895, -1209439895944, 10435153277620, 101078662547567, -892827447251575, -9834570608359487, 88900938146195601, 1101567283699652888

Offset: 0

Ilya Gutkovskiy, Jun 08 2018

sign

			exp(2*x/(exp(x) + 1)) = 1 + x - 2*x^3/3! - x^4/4! + 11*x^5/5! + 13*x^6/6! - 111*x^7/7! - 220*x^8/8! + ...

Wikipedia, Genocchi number
Index entries for sequences related to Bernoulli numbers

Cf. A036968, A296835, A296836.

a:=series(exp(2*x/(exp(x)+1)),x=0,27): seq(n!*coeff(a,x,n),n=0..26); # Paolo P. Lava, Mar 26 2019

nmax = 26; CoefficientList[Series[Exp[2 x/(Exp[x] + 1)], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = Sum[k EulerE[k - 1, 0] Binomial[n - 1, k - 1] a[n - k], {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 26}]
a[n_] := a[n] = Sum[2 (1 - 2^k) BernoulliB[k] Binomial[n - 1, k - 1] a[n - k], {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 26}]

A305922 Expansion of e.g.f. log(1 + 2*x/(exp(x) + 1)).

Original entry on oeis.org

0, 1, -2, 5, -20, 109, -738, 5991, -56760, 614601, -7486670, 101330635, -1508641140, 24503026989, -431137315434, 8169513007215, -165859346028656, 3591802533860497, -82644488286784326, 2013441061219406739, -51777972823724776620, 1401611202556240950645, -39838169568923591411810

Offset: 0

Ilya Gutkovskiy, Jun 14 2018

sign

Logarithmic transform of A036968.

			E.g.f.: A(x) = x - 2*x^2/2! + 5*x^3/3! - 20*x^4/4! + 109*x^5/5! - 738*x^6/6! + ...

Alois P. Heinz, Table of n, a(n) for n = 0..430
N. J. A. Sloane, Transforms
Wikipedia, Genocchi number
Index entries for sequences related to Bernoulli numbers

Cf. A036968, A296837, A296838, A305711.

a:= proc(n) option remember; (t-> `if`(n=0, 0, t(n)-add(a(j)*j*
      t(n-j)*binomial(n, j), j=1..n-1)/n))(i-> i*euler(i-1, 0))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Dec 04 2018

nmax = 22; CoefficientList[Series[Log[1 + 2 x/(Exp[x] + 1)], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = n EulerE[n - 1, 0] - Sum[k Binomial[n, k] (n - k) EulerE[n - k - 1, 0] a[k], {k, 1, n - 1}]/n; a[0] = 0; Table[a[n], {n, 0, 22}]
a[n_] := a[n] = 2 (1 - 2^n) BernoulliB[n] - Sum[k Binomial[n, k] 2 (1 - 2^(n - k)) BernoulliB[n - k] a[k], {k, 1, n - 1}]/n; a[0] = 0; Table[a[n], {n, 0, 22}]

A321595 Decimal expansion of the sum of the reciprocal of the Genocchi numbers (of first kind) with even index (negated).

Original entry on oeis.org

2, 8, 0, 5, 0, 4, 1, 4, 1, 7, 2, 5, 6, 5, 9, 8, 7, 8, 7, 7, 6, 8, 8, 7, 9, 4, 3, 1, 6, 3, 9, 9, 3, 3, 1, 7, 5, 8, 8, 5, 9, 6, 0, 4, 2, 3, 7, 5, 2, 6, 5, 9, 3, 9, 6, 5, 8, 2, 7, 8, 7, 0, 2, 7, 7, 8, 3, 8, 8, 3, 0, 4, 9, 1, 8, 2, 0, 4, 4, 8, 3, 9, 4, 6, 1, 6, 8, 7

Offset: 0

Paolo P. Lava, Nov 14 2018

If also the reciprocal of G(1) = 1 is added we get .7194958582743401212231...

			-.28050414170674716801892095479922438557021976395598927505615374848326107452...

Cf. A036968, A300220.

with(numtheory): P:=proc(h) local a,x,k,n; a:=[seq(factorial(n)*coeff(series(2*x/(1+exp(x)),x=0,h+1),x,n),n=1..h)]; print(evalf(add(1/a[2*k],k=1..trunc(h/2)),200)); end: P(300);

Equals Sum_{k>0} 1/A036968(2*k).

A130653 Odd terms in A002430 = numerators in Taylor series for tan(x).

Original entry on oeis.org

1, 1, 17, 929569, 129848163681107301953, 7724760729208487305545342963324697288405380586579904269441, 357302767470032900576643605538835088084055212588960920085261795996340330997333306469144562500392344758421560010463942134842407723273904635849262137252097

Offset: 1

Alexander Adamchuk, Jun 20 2007

nonn

Odd terms in A002430(n) correspond to the indices that are the powers of 2.

			tan(x) = x + 2 x^3/3! + 16 x^5/5! + 272 x^7/7! + ... = 1*x + 1/3*x^3 + 2/15*x^5 + 17/315*x^7 + 62/2835*x^9 + O(x^10).
A002430(n) begins {1, 1, 2, 17, 62, 1382, 21844, 929569, 6404582, 443861162, 18888466084, 113927491862, 58870668456604, 8374643517010684, 689005380505609448, 129848163681107301953, ...}.
Thus a(1) = 1, a(2) = 1, a(3) = 17, a(4) = 929569, a(5) = 129848163681107301953.

Eric Weisstein's World of Mathematics, Tangent.

Cf. A002430 = Numerators in Taylor series for tan(x). Also from Taylor series for tanh(x). Cf. A001469, A002425, A046990, A089171, A110501, A036968.

Table[ Numerator[ Abs[ 2^(2^n)(2^(2^n)-1)/(2^n)! * BernoulliB[ 2^n ] ] ], {n,1,8} ]

a(n) = Numerator[ Abs[ 2^(2^n)(2^(2^n)-1)/(2^n)! * BernoulliB[ 2^n ] ] ]. a(n) = A002430(2^(n-1)).

cp's OEIS Frontend

A239275 a(n) = numerator(2^n * Bernoulli(n, 1)).

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A227608 Denominators of A225825(n) difference table written by antidiagonals.

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A321217 Genocchi irregular primes.

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A357240 Expansion of e.g.f. 2 * (exp(x) - 1) / (exp(exp(x) - 1) + 1).

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A278331 Shifted sequence of second differences of Genocchi numbers.

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A290701 a(n) = Sum_{k=1..n-1} binomial(n, k)*G_n*G_{n-k} where G_n is the n-th Genocchi number (of the first kind).

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A305711 Expansion of e.g.f. exp(2*x/(exp(x) + 1)).

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A305922 Expansion of e.g.f. log(1 + 2*x/(exp(x) + 1)).

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A290701 a(n) = Sum_{k=1..n-1} binomial(n, k)G_nG_{n-k} where G_n is the n-th Genocchi number (of the first kind).